Problem: The lifespans of tigers in a particular zoo are normally distributed. The average tiger lives $24$ years; the standard deviation is $3.9$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a tiger living longer than $35.7$ years.
Solution: $24$ $20.1$ $27.9$ $16.2$ $31.8$ $12.3$ $35.7$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $24$ years. We know the standard deviation is $3.9$ years, so one standard deviation below the mean is $20.1$ years and one standard deviation above the mean is $27.9$ years. Two standard deviations below the mean is $16.2$ years and two standard deviations above the mean is $31.8$ years. Three standard deviations below the mean is $12.3$ years and three standard deviations above the mean is $35.7$ years. We are interested in the probability of a tiger living longer than $35.7$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the tigers will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the tigers will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $12.3$ years and the other half $({0.15\%})$ will live longer than $35.7$ years. The probability of a particular tiger living longer than $35.7$ years is ${0.15\%}$.